Hamlet's delayed revenge is the entire premise of the play itself. If he had enacted revenge upon his uncle immediately, then the play wouldn't exist. However, because he says he wants to confirm that his uncle is actually the killer first, he delays. Then, when he knows that his uncle did it, and has the chance to kill him while praying, he doesn't, saying he wants to wait until his uncle is doing something evil, to gurantee his soul goes to hell once dead. Interesting reasoning, which could be passed off as cowardice, but, nonetheless, Hamlet does delay.
- His delay sets off a huge chain reaction that leads mostly to the death of everyone that he knows and loves.
- His delay allowed Claudius to scheme with Laertes about killing Hamlet; that scheming led to Laertes, the queen, Claudius AND Hamlet all dying. If Hamlet had killed his uncle before this point, the entire poison scheme wouldn't have arisen, and they would have lived.
- His delay made Claudius suspicious; hence, he and Polonius use Ophelia as a spy to figure out what he is up to. This manipulation leads to the further alienation of his relationship with Ophelia, plunging her further into depression.
These reactions are all the major events of the play, all caused by his delay to enact revenge. I hope that helped; good luck!